The value of PI in Cheops' Pyramid
PI in the side view
We now consider the side view,
or elevation, of the Great Pyramid.
The height of the Great Pyramid is given
as 481.4 feet by Edwards, p. 97.
Thus from due North (or East, etc.) of
the pyramid, on the desert, we
see an isosceles triangle of width
about 756 feet, and height (if its
top were intact) of 481.4 feet.
The angles at the lower corners are about 52 degrees.
In the middle of p. 254 Edwards mentions
this trigonometric fact, apparently first
noticed John Taylor around 1837 (see Tompkins, p. 70):
The normal angle of incline was
about 52 degrees - a slope which,
in the pyramid of Meidum and in the
Great Pyramid, would have resulted if the height had been
made to correspond with the radius of a circle
the circumference of which was equal
to the perimeter of the pyramid at ground level.
In other words,
- a circle with radius r has circumference 2*Pi*r,
- a pyramid with square base of width w
has perimeter 4*w
- if the radius, r, is equal to the height
of the pyramid, h, and the circumference
of the circle is equal to the perimeter of
the pyramid, then 2*Pi*h = 4*w
or h/w = 4/2*Pi = 2/Pi.
- the angle of incline, or slope, is the arctan
of the ratio of height to half the width,
or h/(w/2) = 2*h/w = 4/Pi.
This angle is 51 degrees 52 minutes, very close to 52 degrees.
In Cheop's pyramid, the accuracy of this angle is within
one part in 1000, according to Mendelssohn, p. 50.
See for yourself:
here is a list of ratios, h/w.
- 2/Pi: 0.63662 (Pi = 3.14159, our value)
- 2/Pi: 0.628906 (Pi = 256/81, ancient Egyptian value)
- North face: 0.63725
- South face: 0.6367
- East face: 0.6369
- West face: 0.6370
All of these ratios measured by Cole for the Great Pyramid
are closer to 2/Pi-real (better than one part
in 1000) than to 2/Pi-Egyptian.
This is what Mendelssohn calls "the unsolved problem"
of Cheop's pyramid.
For this problem he offers a tentative solution,
attributed (on p. 64) to an engineer, T. E. Connolly.
This is the rolled cubit theory on p. 73.
The astonishing accuracy with which the ratio
of height to circumference of
the Great Pyramid represents the squaring of the circle
1/2*Pi was possibly due to the fact that the Egyptians
may have measured long horizontal distances by counting the revolutions
of a rolling drum. In this way they would have
arived at the transcendental number Pi...
without realizing it.
We suggest you try this out:
roll a drum one cubit (20.62 inches) in diameter
a distance of about three city blocks (775 feet,
or about 150 revolutions)
and obtain an accuracy
in the measured distance better than one part in 1000,
that is, 8 inches.
Even though this seems unlikely, we must nevertheless
admit the possible of an accurate unit of measure,
the rolled cubit, of about 65.78 inches.
Back to Brunés' construction #1
Recall,
that his Fig. 83 fits the pyramid elevation into
the top half of the circle's rectangle. If this
fit were good, the ratio of height to width would
be 5:8, or 0.6250: too small! Hence the sloppiness
of that fit: the pyramid was a bit too narrow
for the circle's rectangle.
PI elsewhere in the Great Pyramid
Petrie discovered that the King's Chamber walls
were proportioned like the elevation: length
over side-wall perimeter as 1 is to PI. See Tompkins, p. 101.
References
- Peter Tompkins, Secrets of the Great Pyramid, 1971.
- The penultimate pyramidiot text.
Ralph H. Abraham, 29 April 1996.